Over the past several years, there has been considerable interest in using optical scatterometry (i.e., optical diffraction) to perform critical dimension (CD) measurements of the lines and structures included in integrated circuits. Optical scatterometry has been used to analyze periodic two-dimensional structures (e.g., line gratings) as well as three-dimensional structures (e.g., patterns of vias or mesas). Scatterometry is also used to perform overlay registration measurements. Overlay measurements attempt to measure the degree of alignment between successive lithographic mask layers.
Various optical techniques have been used to perform optical scatterometry. These techniques include broadband scatterometry (U.S. Pat. Nos. 5,607,800; 5,867,276 and 5,963,329), spectral ellipsometry (U.S. Pat. No. 5,739,909) as well as spectral and single-wavelength beam profile reflectance and beam profile ellipsometry (U.S. Pat. No. 6,429,943). In addition it may be possible to employ single-wavelength laser BPR or BPE to obtain CD measurements on isolated lines or isolated vias and mesas.
Most scatterometry systems use a modeling approach to transform scatterometry signals into critical dimension measurements. For this type of approach, a theoretical model is defined for each physical structure that will be analyzed. A series of calculations are then performed to predict the empirical measurements (optical diffraction) that scatterometry systems would record for the structure. The theoretical results of this calculation are then compared to the measured data (actually, the normalized data). To the extent the results do not match, the theoretical model is modified and the theoretical data is calculated once again and compared to the empirical measurements. This process is repeated iteratively until the correspondence between the calculated theoretical data and the empirical measurements reaches an acceptable level of fitness. At this point, the characteristics of the theoretical model and the physical structure should be very similar.
The most common technique for calculating optical diffraction for scatterometry models is known as rigorous coupled wave analysis, or RCWA. For RCWA, the diffraction associated with a model is calculated by finding solutions to Maxwell's equations for: 1) the incident electromagnetic field, the electromagnetic field within the model, and 3) the output or resulting electromagnetic field. The solutions are obtained by expanding the fields in terms of exact solutions in each region. The coefficients are obtained by requiring that the transverse electric and magnetic fields be continuous (modal matching). Unfortunately, the exact solutions require repeated matrix diagonalizations in each region in addition to few other matrix operations. Computationally, matrix diagonalization is exceedingly slow, generally more than ten times slower than other matrix operations such as matrix-matrix multiplications or matrix inversions. As a result, RCWA tends to be slow, especially for materials with complex dielectric constants. As the models become more complex (particularly as the profiles of the walls of the features become more complex) the calculations become exceedingly long and complex. Even with high-speed processors, real time evaluation of these calculations can be difficult. Analysis on a real time basis is very desirable so that manufacturers can immediately determine when a process is not operating correctly. The need is becoming more acute as the industry moves towards integrated metrology solutions wherein the metrology hardware is integrated directly with the process hardware.
A number of approaches have been developed to overcome the calculation bottleneck associated with the analysis of scatterometry results. Many of these approaches have involved techniques for improving calculation throughput, such as parallel processing techniques. An approach of this type is described in U.S. Pat. No. 6,704,661, (incorporated herein by reference) which describes distribution of scatterometry calculations among a group of parallel processors. In the preferred embodiment, the processor configuration includes a master processor and a plurality of slave processors. The master processor handles the control and the comparison functions. The calculation of the response of the theoretical sample to the interaction with the optical probe radiation is distributed by the master processor to itself and the slave processors.
For example, where the data is taken as a function of wavelength, the calculations are distributed as a function of wavelength. Thus, a first slave processor will use Maxwell's equations to determine the expected intensity of light at selected ones of the measured wavelengths scattered from a given theoretical model. The other slave processors will carry out the same calculations at different wavelengths. Assuming there are five processors (one master and four slaves) and fifty wavelengths, each processor will perform ten such calculations per iteration.
Once the calculations are complete, the master processor performs the best-fit comparison between each of the calculated intensities and the measured normalized intensities. Based on this fit, the master processor will modify the parameters of the model as discussed above (changing the widths or layer thickness). The master processor will then distribute the calculations for the modified model to the slave processors. This sequence is repeated until a good fit is achieved.
This distributed processing approach can also be used with multiple angle of incidence information. In this situation, the calculations at each of the different angles of incidence can be distributed to the slave processor. Techniques of this type are an effective method for reducing the time required for scatterometry calculations. At the same time, the speedup provided by parallel processing is strictly dependent on the availability (and associated cost) of multiple processors. Amdahl's law also limits the amount of speedup available by parallel processing since serial program portions are not improved. At the present time, neither cost nor ultimate speed improvement is a serious limitation for parallel processing techniques. As geometries continue to shrink, however it becomes increasingly possible that computational complexity will outstrip the use of parallel techniques alone.
Another approach is to use pre-computed libraries of predicted measurements. This type of approach is discussed in (U.S. Pat. No. 6,483,580) as well as the references cited therein. In this approach, the theoretical model is parameterized to allow the characteristics of the physical structure to be varied. The parameters are varied over a predetermined range and the theoretical result for each variation to the physical structure is calculated to define a library of solutions. When the empirical measurements are obtained, the library is searched to find the best fit.
The use of libraries speeds up the analysis process by allowing theoretical results to be computed once and reused many times. At the same time, library use does not completely solve the calculation bottleneck. Construction of libraries is time consuming, requiring repeated evaluation of the same time consuming theoretical models. Process changes and other variables may require periodic library modification or replacement at the cost of still more calculations. For these reasons, libraries are expensive (in computational terms) to build and to maintain. Libraries are also necessarily limited in their resolution and can contain only a finite number of theoretical results. As a result, there are many cases where empirical measurements do not have exact library matches. One approach for dealing with this problem is to generate additional theoretical results in real time to augment the theoretical results already present in the library. This combined approach improves accuracy, but slows the scatterometry process as theoretical models are evaluated in real time. A similar approach is to use a library as a starting point and apply an interpolation approach to generate missing results. This approach avoids the penalty associated with generating results in real time, but sacrifices accuracy during the interpolation process. See U.S. Pat. No. 6,768,967, incorporated herein by reference.
For these reasons and others, there is a continuing need for faster methods for computing theoretical results for scatterometry systems. This is true both for systems that use real time evaluation of theoretical models as well as systems that use library based problem solving. The need for faster evaluation methods will almost certainly increase as models become more detailed to more accurately reflect physical structures.